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Search: id:A128727
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| A128727 |
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Triangle read by rows: T(n,k) is the number of skew Dyck paths of semilength n having k DDU and LDU's. A skew Dyck path is a path in the first quadrant which begins at the origin, ends on the x-axis, consists of steps U=(1,1)(up), D=(1,-1)(down),and L=(-1,-1)(left) so that up and left steps do not overlap. The length of a path is defined to be the number of steps in it. |
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1
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| 1, 1, 3, 9, 1, 27, 9, 81, 54, 2, 243, 270, 30, 729, 1215, 270, 5, 2187, 5103, 1890, 105, 6561, 20412, 11340, 1260, 14, 19683, 78732, 61236, 11340, 378, 59049, 295245, 306180, 85050, 5670, 42, 177147, 1082565, 1443420, 561330, 62370, 1386, 531441
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OFFSET
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0,3
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COMMENT
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Row n has ceil(n/2) terms (n>=1). Row sums yield A002212. T(n,0)=3^(n-1). T(2k+1,k)=binom(2k,k)/(k+1) (the Catalan numbers, A000108). T(2k,k-1)=3binom(2k-1,k)=A003409(k). Sum(k*T(n,k),k>=0)=A026377(n-1).
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REFERENCES
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E. Deutsch, E. Munarini, and S. Rinaldi, Skew Dyck paths (in preparation).
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FORMULA
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T(n,k)=(1/n)*3^(n-1-2k)*binom(n,k)*binom(n-k,k+1) G.f.=G=G(t,z) satisfies tzG^2-(1-3z+2tz)G+1-2z+tz=0.
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EXAMPLE
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T(5,2)=2 because we have UU(DDU)U(DDU)D and UUU(DDU)(DDU)D (the 2 subwords are shown between parentheses).
Triangle starts:
1;
1;
3;
9,1;
27,9;
81,54,2;
243,270,30;
729,1215,270,5;
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MAPLE
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T:=(n, k)->3^(n-1-2*k)*binomial(n, k)*binomial(n-k, k+1)/n: 1; for n from 1 to 13 do seq(T(n, k), k=0..floor((n-1)/2)) od; # yields sequence in triangular form
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CROSSREFS
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Cf. A002212, A000108, A003409, A026377.
Adjacent sequences: A128724 A128725 A128726 this_sequence A128728 A128729 A128730
Sequence in context: A019817 A080322 A126179 this_sequence A126177 A128733 A128724
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KEYWORD
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nonn,tabf
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AUTHOR
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Emeric Deutsch (deutsch(AT)duke.poly.edu), Mar 31 2007
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